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State-multiplicative Noisy Systems: Robust H∞ Output-Feedback Control
The International Federation of Congress Automatic Control Proceedings of the 20th World Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 3823–3828 State-multiplicative Noisy Systems: Robust State-multiplicative Noisy H∞ Output-Feedback Control Robust State-multiplicative Noisy Systems: Systems: Robust H Output-Feedback Control ∞ H∞ Output-Feedback Control ∗,∗∗ ∗∗
Eli Gershon Uri Shaked ∗∗ Eli Gershon ∗,∗∗ ∗,∗∗ Uri Shaked ∗∗ Eli Gershon Uri Shaked ∗ Holon Institute of Technology, Medical Engineering, Holon, Israel ∗ ∗∗ Institute of Technology, Medical Engineering, Holon, Israel School of Electrical Engineering, Tel-Aviv University, Tel-Aviv ∗ Holon Holon Institute ofIsrael Technology, Medical Engineering, Holon, Israel ∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, (e-mail: gershon@ eng.tau.ac.il) ∗∗ School 69978, of Electrical Engineering, Tel-Aviv University, Tel-Aviv Israel (e-mail: gershon@ eng.tau.ac.il) 69978, Israel (e-mail: gershon@ eng.tau.ac.il) Abstract: Linear discrete-time systems with stochastic and deterministic polytopic type Abstract: systems stochastic and deterministic polytopic type uncertaintiesLinear in theirdiscrete-time state-space model arewith considered. A dynamic output-feedback controller Abstract: Linear discrete-time systems stochastic and deterministic polytopic type uncertainties in atheir state-space model arewith considered. A of dynamic output-feedback controller is obtained via new approach which allows a derivation a controller in spite of parameter uncertainties in their state-space model are considered. A dynamic output-feedback controller is obtained via newproposed approachapproach which allows derivation of a controller spite of parameter uncertainty. In athe the asystem is described via aindifference equation is obtained via new approach which asystem derivation of a controller spite ofparameters. parameter uncertainty. In athe proposed approach described via controller aindifference equation and an augmented system is then usedallows tothe obtain the isoutput-feedback uncertainty. In isthe proposed approach system described tovia a quadratic difference equation and augmented system is then used tothe obtain the isoutput-feedback parameters. The an controller obtained without assuming a specific structure thecontroller Lyapunov and an augmented system is then to obtain the output-feedback parameters. The controller obtained without assuming a specific structure is toobtained thecontroller quadratic Lyapunov function and it is the first time thatused an output-feedback controller for robust stateThe controller is obtained without a the specific structure toobtained the quadratic Lyapunov function and itsystems. is the first that assuming an output-feedback controller is for robust statemultiplicative Thetime controller minimizes stochastic l2 −gain of the closed-loop where function and itsystems. is defined the first thatexpected an output-feedback is obtained for robust statemultiplicative The controller minimizes theofstochastic l2 −gain the closed-loop a cost function is totime be the value the controller standard H performance indexwhere with ∞ of multiplicative systems. The controller minimizes the stochastic l −gain of the closed-loop where 2 a cost function is defined uncertainty. to be the expected value of standard H∞ performance indexofwith respect to the stochastic An example is the given that demonstrates the merit the a cost function is defined uncertainty. to be the expected value of standard H∞ performance indexofwith respect to the stochastic An example is the given that demonstrates the merit the theory. respect theory. to the stochastic uncertainty. An example is given that demonstrates the merit of the theory. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Stochastic H∞ output-feedback control, vertex-dependent Lyapunov function, Keywords: Stochastic multiplicative noise. H∞ output-feedback control, vertex-dependent Lyapunov function, Keywords: Stochastic multiplicative noise. H∞ output-feedback control, vertex-dependent Lyapunov function, multiplicative noise. 1. INTRODUCTION shon and Shaked [2008]. The solution in Dragan and Stoica 1. INTRODUCTION shon Shakedthe [2008]. Theand solution in Dragan andhorizon Stoica [1998]and includes finite the infinite-time 1. INTRODUCTION shon and Shaked [2008]. Theand solution in Dragan andhorizon Stoica [1998] includes the finite the infinite-time problems without transients. One drawback of Dragan and The field of stochastic H∞ control and estimation of state[1998] includes the finite and the infinite-time horizon problems without transients. One drawback of Dragan and Stoica [1998] is the fact that in the infinite-time horizon The field of stochastic control and estimation of statemultiplicative systemsHhas been developed following the ∞ problems without transients. One drawback of Dragan and control andtheory estimation stateThe fieldin of the stochastic Hhas ∞H is number the factofthat in the infinite-time horizon case, an[1998] infinite Linear Matrix Inequality (LMI) multiplicative systems been developed following the advance stochastic that of was de- Stoica 2 control Stoica [1998] is the fact that in the infinite-time horizon multiplicative systems has been developed following the case, an infinite number of Linear Matrix Inequality (LMI) sets should be solved. We also note that the fact that advance in inthe theearly stochastic H270’s control theory that developed 60’s and (see Gershon et al.was [2005] an infinite of Linear Matrix Inequality (LMI) advance inthe theearly stochastic H270’s theory that de- case, sets should be number solved. We also that the fact that in Dragan and Stoica [1998] thenote measurement coupling veloped in 60’s and (seeemerged, Gershon et aal.was [2005] for a comprehensive study). Itcontrol has as natural sets should be solved. We also note that the fact that veloped in the early 60’s and 70’s (see Gershon et al. [2005] Dragan andaccommodate Stoica [1998]any theuncertainty measurement coupling matrix can not is a practical for a comprehensive It progress has emerged, natural extension, in parallelstudy). with the madeastoathe field in in Dragan and Stoica [1998] the measurement coupling for a comprehensive study). It has emerged, as a natural can not anywhere uncertainty is a practical restriction, for accommodate example in cases the measurements extension, in parallel with the progress made field matrix of deterministic H∞ control theory, drawing onto thethe advent not accommodate any uncertainty is a practical extension, in field parallel with the progress made the field matrix restriction, forderivatives example in(e.g. cases where the measurements include can state acceleration control of an deterministic H∞tocontrol theory, drawing onto the advent of the latter cope with MIMO control problems restriction, for example in cases where the measurements of deterministic H control theory, drawing on the advent derivatives acceleration control et of al. an aircraft state or missile). The (e.g. treatment of Bouhtouri of the and latterLimebeer field ∞to cope control Green [1995]with and MIMO with the robustproblems control include include state derivatives acceleration control of al. an of latterLimebeer field to cope with control problems or missile). The (e.g. treatment Bouhtouri et [1999] includes the derivation of the of stochastic Bounded Green and [1995] and MIMO withWeiland the robust control andthe estimation problems, Scherer and [2006]. This aircraft aircraft or missile). The treatment of Bouhtouri et al. Green and Limebeer [1995] and with the robust control includes the derivation of the Bounded Real Lemma (BRL) and concerns onlystochastic the stationary case and problems, and Weiland fieldestimation is recognized now asScherer a central sub-field [2006]. withinThis the [1999] includes the nonlinear derivation of the Bounded and estimation and Weiland [2006]. This Real (BRL) and concerns onlystochastic thewere stationary case whereLemma two coupled inequalities obtained. field is of recognized now asScherer acontrol central sub-field within the [1999] theory linearproblems, stochastic Dragan and Morozan Real Lemma (BRL) and concerns only the stationary case field is of recognized now as[2004] acontrol central sub-field theory linear andwithin Morozan [1997], Chen andstochastic Zhang (seeDragan also Gershon et the al. where two coupled nonlinear inequalities were obtained. In Gershon and Shaked [2008] the solution of the outputtheory of control Dragan and Morozan [1997], Chen andstochastic Zhang [2004] (see also Gershon et al. where two coupled nonlinear inequalities were obtained. [2005] for alinear comprehensive review) and nonlinear stochasIn Gershon and Shaked the obtained solution of outputfeedback control problem[2008] has been bythe transform[1997], Chen and Zhang [2004] (see also Gershon et al. [2005] for a (see, comprehensive review) nonlinear stochastic systems for example, Dongand et al. [2015],Hu et al. In Gershon and to Shaked [2008] theInobtained solution of the outputfeedback control problem has been by transforming the problem one of filtering. the finite horizon case [2005] for a (see, comprehensive review) nonlinear stochastic systems forsolution example, Dongand et al. [2015],Hu et al. [2013]). Numerous methods have been applied to feedback control problem has been obtained by transforming the problem oneobtained of filtering. In via theDLMIs finite horizon case the solution has to been there [Difference tic systems (see, for example, Dong et al. [2015],Hu et al. [2013]). Numerous methods have applied to ing the problem to one of filtering. In the finite horizon case various control and solution estimation patterns (seebeen Gershon et al. the solution hasInequalities] been obtained there via DLMIs [Difference Linear Matrix whereas in the stationary case [2013]). Numerous solution methods have been applied to various control and estimation patterns (seethe Gershon et al. the solution has been obtained there via DLMIs [Difference [2005] and the references therein) and over last decade Matrix whereas in the stationary case the solution hasInequalities] been obtained by assigning a special strucvarious control and patterns (see Gershon et al. Linear [2005] and the references andtoover the last decade the latter field has estimation been therein) expanded include stochastic Linear Matrix Inequalities] whereas inthereby the stationary case the solution has been obtained by assigning a special structure to the Lyapunov matrix solution, imposing an [2005] and thetime references last decade the latter field hasdelay beenoftherein) expanded toover include stochastic systems with various and kinds (i.ethe constant time- the solution has been obtained by assigning a special structure to the Lyapunov matrix The solution, thereby imposing over-design on the solution. robust case could not an be the latter field has been expanded to include stochastic systemsslow withand timefast delay of various kinds time- ture to the Lyapunov matrix solution, thereby imposing an delay, varying delay) in (i.e bothconstant the nominal over-design on the solution. The robust case could notfact be treated in Gershon and Shaked [2008] owing to the systems with time delay of various kinds (i.e constant timedelay, and fast varying delay) inand both the nominal and theslow uncertain cases (see Gershon Shaked [2013], over-design on obtained the solution. The robust caseincould notfact be treated in Gershon and Shaked [2008] owing to the that the LMI there is not affine the system delay, and Gao fast varying delay) both the nominal and theslow uncertain cases (see and Gershon and Shaked [2013], see also Li and [2011a] Li in and Gao [2011b] for treated in Gershon and Shaked [2008] owing to the fact that the LMI is not an affine in thedynamic system matrices. The obtained problem there of finding optimal and the uncertain cases (see and Gershon andGao Shaked [2013], see Li and Gao [2011a] Li and [2011b] for that the LMI obtained there is not affine in the system the also deterministic case). matrices. The problem of finding an optimal output-feedback for state-multiplicative systems dynamic has thus see also Li and Gao [2011a] and Li and Gao [2011b] for the deterministic case). matrices. The problem of finding an optimal dynamic output-feedback for state-multiplicative systems has used thus been constrained in the past by the solution method the deterministic case). output-feedback for state-multiplicative systems has used thus been constrained in the past by the solution method and conservative solutions have then been obtained. The problem of H∞ dynamic output-feedback has always been constrained in the past by the solution method used The problem of H output-feedback has In always been a central issue in modern control theory. the and conservative solutions have then been obtained. ∞ dynamic In present manuscript solve thebeen problem of discreteandthe conservative solutionswe have then obtained. The problem of H output-feedback always ∞ dynamic been a central issue incase, modern control theory. In the stochastic discrete-time this problem was has tackled by In the present manuscript we solve the problem of discretetime dynamic output-feedback control via a new approach been central issue inDragan modern control theory. InGerthe stochastic discrete-time case, this was tackled by In the present manuscript we solve the problem of discretevariousa research groups andproblem Stoica [1998] and time dynamicusoutput-feedback via athe newproblem approach that enables to solve, for thecontrol first time, of stochastic discrete-time case, this problem was tackled by various research groups Dragan and Stoica [1998] and Ger- time dynamic output-feedback control via a new approach that enables us to solve, for the first time, the problem of measurement control of uncertain systems without the re various research groups Dragan and Stoica [1998] and GerThis work was supported by the Israeli Science Foundation no. that enables to solve, for the first the problem of measurement control of uncertain systems without the restrictions thatus have been indicated in time, the above mentioned 737/2011. This work was supported by the Israeli Science Foundation no. measurement control of uncertain systems without the re This work was supported by the Israeli Science Foundation no. strictions that have been indicated in the above mentioned 737/2011. strictions that have been indicated in the above mentioned 737/2011.
Copyright © 2017 IFAC 3886 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2017 responsibility IFAC 3886Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 3886 10.1016/j.ifacol.2017.08.488
Proceedings of the 20th IFAC World Congress 3824 Eli Gershon et al. / IFAC PapersOnLine 50-1 (2017) 3823–3828 Toulouse, France, July 9-14, 2017
works. Applying a linear fractional approach, we apply the method of Suplin and Shaked [2005] and transform the problem of finding a dynamic controller to one of finding a state-feedback controller gain matrix. We are thus able to treat two control design cases which are fundamental in control engineering on one hand and, on the other hand, we are also able to cope with the robust version of the latter two control patterns, without pre-assigning structure to the relevant Lyapunov matrix function solution. In the uncertain case, we bring a vertex-dependent solution (in contrast to the traditional ‘quadratic‘ solution). The paper is organized as follows: starting with the difference equation representation, we first introduce in Section 2 the linear fractional intermediate description of the system. In Section 3, we consider the stabilization and the l2 −gain of the nominal closed-loop system. Based on the result of Section 3, the solution of the robust dynamic output-feedback control problem is given in Section 4 for the case where the system encounters polytopictype parameter uncertainty. Two different control solution methods are applied there: the quadratic and the vertex dependent approaches. In Section 5, an example is given that demonstrates the applicability and tractability of our method we developed. Notation: Throughout the paper the superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space, Rn×m is the set of all n × m real matrices, N is the set of natural numbers and the notation P > 0, (respectively, P ≥ 0) for P ∈ Rn×n means that P is symmetric and positive definite (respectively, semidefinite). We denote by L2 (Ω, Rn ) the space of squareintegrable Rn − valued functions on the probability space (Ω, F, P), where Ω is the sample space, F is a σ algebra of a subset of Ω called events and P is the probability measure on F . By (Fk )k∈N we denote an increasing family of σ-algebras Fk ⊂ F. We also denote by ˜l2 (N ; Rn ) the ndimensional space of nonanticipative stochastic processes {fk }k∈N with respect to (Fk )k∈N where fk ∈ L2 (Ω, Rn ). On the latter space the following l2 -norm is defined: ∞ ∞ ||fk ||2 } = E{||fk ||2 } < ∞, ||{fk }||˜2l = E{
where
¯ i = 1, ..., L,
(3)
¯ is the number of vertices. and where L We seek an output-feedback dynamical controller of the type l−2 l−1 ¯l−1−i yk+i uk+l−1 + (4) A¯i uk+i = B i=0
i=0
that, over the polytope Ω stabilizes the system exponentially in the mean-square sense and minimizes the induced l2 −norm of the resulting closed-loop.
The above system is described (for clarity of representation we take Fi = 0, i = l, ..., 2l − 1) in the block diagram of Figure 1 where δ denotes the one unit time shift operator. We represent the above system by the augmented observability canonical form (see Kailath [1980]) as follows: we ∆ define the following augmented state-vector in n , n = l(m + r) − r, T T T (5) ξk = yk+l−1 . . . ykT uTk+l−2 . . . uTk , yk+l−2
and obtain ˜ k + νk F˜ ξk + (B ˜ + νk B)Kξ ˆ ξk+1 = Aξ k where
2
0
¯l−1, i ¯0, i ... N N ¯ l−1, i , ¯ 0, i ... D Ωi = D F0, i ... F2l−2, i
0
{fk } ∈ ˜l2 (N ; Rn ),
where || · || is the standard Euclidean norm.
−D¯ l−1
˜ A=
Im 0 . 0 0 0 0 . 0
−Fl−1
F˜ =
0 0 . 0 0 0 0 . 0
. 0 Im . . . . . . . . 0 0 . . . . . . .
¯0 N ¯l−1 ¯1 . . . N . . −D . . 0 0 . . . 0 . . 0 0 . . . 0 . . . . . . . . . Im 0 0 . . . 0 , . . 0 0 . . . 0 . . 0 Ir 0 . . 0 . . 0 0 Ir . . 0 . . . . . . . . . . 0 0 . . Ir 0
. . . . . . . . . .
. −F0 Fl+1 . 0 0 . 0 0 . . . 0 0 0 . 0 0 . 0 0 . 0 0 . . . . 0 0
. . . . . . 0 0 . .
. . . . . . . . . .
(7)
. F2l−1 . 0 . 0 . . . 0 , . 0 . 0 . 0 . . 0 0
T ˆ = F T 0. . . 0 0 T ˜= N ¯0 0. . . 0 Ir 0 . . . 0 T , B B l
2. PROBLEM FORMULATION
(6)
(7) We consider the r-inputs m-outputs system that is de 2l−1 ] = and K=[K1 K2 , ..., Kl Kl+1 , ..., K scribed by the following difference equation: ¯0 B ¯l−1−A¯l−2 −A¯l−3 . . .−A¯0 . ¯1. . .B B l−1 l−1 ¯ i + Fi νk )yk+i = ¯l−1−i +F2l−1−i νk )uk+i (1) We note that the above state space model is non minimal, (D (N yk+l + for both the nominal and the uncertain cases. It is already i=0 i=0 observable but may be uncontrollable due to the additional m r where yk ∈ R is the system output, uk ∈ R is the roots added to the augmented system. A ’state-feedback’ control input, {νk } is a standard zero-mean real scalar from the state vector ξ in this model to the ’control input’ k white-noise sequence with E{νk νj } = δkj and where u will produce the components Ai and Bi of the k+l−1 ¯i ∈ Rm×r , Fi ∈ Rm×r , i = l, ..., 2l − ¯ i ∈ Rm×m , N D controller (4) that relates y k with uk . 1 and Fi ∈ Rm×m , i = l, ..., l − 1. We assume that system is detectable and stabilizable. We further assume that the In the stochastic H∞ control setting, we obtain the followparameters in (1) are not precisely known and that they ing state-space representation: ˜ k+B ˜w wk + BKξ ˜ k + νk (F˜ + BK)ξ ˆ reside in the following polytope, ξk+1 = Aξ k, (8) ˜ 11 wk , ˜ ξk + D Ω = Co{Ω1 , Ω2 , ..., ΩL¯ } , (2) zk = C˜1 ξk + νk W 3887
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Eli Gershon et al. / IFAC PapersOnLine 50-1 (2017) 3823–3828
˜ and B ˆ depend on the way by which νk affects where W ¯ Ni , i = 1, 2, ..., N. In our model, the disturbance acts on yk+l with
∆ ΓP =
¯1 ...N ¯l−1 ] + N ¯0 K. (9) ˜w = [Im 0 0 ...0]T and C˜1 = [0...0 N B ˜ 11 depends on the control problem to be The matrix D solved. It describes the way that the disturbance affects the objective function. In the following we describe two problems in which the disturbance effect is considered differently: Problem 1: Sensitivity minimization The sensitivity matrix function is the operator that relates the input {wk } in Figure 1 with the signal that is measured at the bullet that immediately proceeds the disturbance entry point. The minimization of the H∞ norm of the ˜ 11 = 0 sensitivity function is thus achieved by taking D in (8b). Problem 2: Complementary sensitivity minimization The complementary sensitivity is the operator that relates the input {wk } in Figure 1 with the signal that precedes the entry point. In our case, we seek to reduce the l2 −gain ˜ 11 = 0. of the latter transference. We thus require that D We note that in our system the measurement signal is not contaminated by sensor noise. However, the complementary sensitivity function is also the transference (noting that the phase is reversed) from the traditional entry point of the measurement noise to the output measurement signal. By minimizing the induced L2 norm of this function one also minimizes the effect of measurement noise if it exists.
3. THE H∞ CONTROLLER FOR NOMINAL SYSTEMS In order to solve the last two minimization problems for the augmented system we first bring the stochastic BRL result that was obtained in (see Gershon et al. [2005], p.104). This result refers to the following n-th order system xk+1 = (A + F νk )xk + B1 wk , zk = (C1 + W νk )xk + D11 wk ,
xi = 0, i ≤ 0,
(10)
0
P AT P F T P C1T P W T
∗ −γ 2 Ip B1T
∗
T D11
0
0
0
0
∗
−P
0
0 0
∗
∗
−Ir
∗
−P
∗
∗
∗
∗
∗
∗
∗
0
∗
∗
−Ir
< 0. (12)
Remark 1: Lemma 1 has been used in the past to solve the output feedback control problem, Gershon et al. [2005], Gershon and Shaked [2008]. Unfortunately, when seeking a dynamical output-feedback controller, the resulting inequalities become BMIs [Bilinear Matrix Inequalities]. To circumvent this difficulty, a special structure had to be assigned to the matrix P - a procedure which led to considerable overdesign. Based on Lemma 1, the ’state-feedback controller’ can be readily derived for the augmented system by applying the latter result to the system of (8a,b). In this way no special assumption on the structure of P is required. We thus obtain the following result: Theorem 1 Consider the augmented system (8a,b) and (11). For a prescribed scalar γ > 0, there exists a statefeedback gain that achieves negative JE for all nonzero 2 wk ∈ ˜lF ([0, ∞); Rp ), if there exist n1 × n1 matrix P > 0, k where n1 = l(m + r) − r and a matrix Y that satisfy the following LMI condition: −P 0 P A˜T + Y T B˜ T P F˜ T + Y T Bˆ T P C˜ T P W ˜T 1
0
˜T D 11
0
−P
0
0
0
∗
∗
−P
0
0
∗
∗
∗
−Ir
0
∗
∗
∗
∗
−Ir
∗
2
−γ Ip
˜T B w
∗
∗
∗ ∗ ∗
< 0.(13)
In the latter case, the ’state-feedback gain’ is given by: ¯0 B ¯l−1 −A¯l−2 −A¯l−3... −A¯0 = Y P −1 , (14) ¯1... B K= B
from where the system matrices of the actual output feedback controller of Figure 2 are obtained. Proof We replace A, B1 and C1 , W, D11 of (10a,b) by ˆ ˜w and C˜1 , W ˜, D ˜ 11 of (8a,b) respectively, and A˜ + BK, B we denote KP by Y. Substituting in (12) the LMI of (13) is readily obtained.
with the index of performance:
4. THE H∞ CONTROLLER FOR UNCERTAIN SYSTEMS
∆
JE = ||zk ||˜2l − γ 2 ||wk ||˜2l , 2
−P
3825
2
(11)
where wk ∈ Rp , zk ∈ Rr . The following has been obtained in Dragan and Stoica [1998],Gershon et al. [2005]: Lemma 1 Consider the system (10a,b) and the above JE . The system is exponentially stable in the mean square sense and, for a prescribed scalar γ > 0, the requirement of 2 JE < 0 is achieved for all nonzero wk ∈ ˜lF ([0, ∞); Rp ), iff k there exists n×n matrix P > 0, that satisfies the following inequality:
The main advantage of our approach is the ability it provides to obtain a solution to the output-feedback control problem in the case of polytopic uncertain parameters. Two possible solution methods can be considered. The first produces a ’quadratic’ solution where a single Lyapunov function is considered over the whole uncertainty interval. This is readily obtained by applying the result of Theorem 1 to all the vertices of the uncertainty polytope applying the same matrices P and Y. The second solution method is based on the fact that a less conservative solution may
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be obtained by considering vertex-dependent Lyapunov functions over the whole uncertainty polytope. In order to apply the latter approach, we first present the vertexdependent solution of the BRL of Lemma 1. We consider the system of (10a,b) and the LMI of (12). Applying Schur’s complements to the latter LMI, we obtain T
where
Ψ + ΦP Φ ≤ 0,
Ψ=
−γ 2 Ip B1T
0
0
∗
−P
0
0
∗
∗
−P
0
∗
∗
∗
−Ir
∗
∗
∗
∗
0
0
A = F . , Φ 0 0 C1 0
−Ir
(15)
¯ ∀j = 1, 2, .., L. Applying the above result, the solution of the vertexdependent robust BRL problem can be readily derived for the augmented system. Considering the augmented system (8a,b) where K = 0 and (11), where the system matrices components lie within the polytope of (2), we obtain the following condition: (j) ˆ +GT Φ(j)T +Φ(j) G −GT +Φ(j) H ∆ Ψ ˆ (j) = < 0, Ω ∗ −H −H T +P (j)
W
The structure of (15) can be used in order to reduce the conservatism entailed by applying the Lyapunov function uniformly over the uncertainty polytope. However, the standard application of the latter method in Oliveira and Skelton [2001] is not readily applicable to the stochastic case where taking the adjoint of the system, instead of the original one, is theoretically unjustified. We therefore bring below a vertex-dependent method that is more suitable and readily applicable to the uncertain stochastic case. We start with (15) and obtain the following lemma: Lemma 2: Inequality (15a) is satisfied iff there exist matrices: 0 < P ∈ Rn×n , G ∈ Rn×(2n+p+r) and H ∈ Rn×n that satisfy the following inequality T T −GT + ΦH ∆ Ψ + G Φ + ΦG Ω= < 0. (16) ∗ −H − H T + P Proof: Substituting G = 0 and H = P in (16), inequality (15a) is obtained. To show that (16) leads to (15a) we consider I 0 ΨΦ(1,1) ΨΦ(1,2) I Φ = , Ω T ∗ ΨΦ(2,2) 0 I Φ I
Φ
¯ ∀j = 1, 2, ..., L,
0 , 0 0
(19)
∗ ∗ ∗ ∗ −Ir (j)T ˜ T ˜ (j)T ˜ T ¯ ˜ = [0 A F C1 W ], ∀j = 1, 2, ...L.
(j)
−γ 2 I
∗
∗
ΓOF < 0, where p
˜T B w
0
0
0
(j) (j) (j) T ˜ (j)T T ˜ T αH W ΓOF (2, 6) ΓOF (2, 2) ΓOF (2, 3) αH C 1
∗
−P
∗
∗ ∗
∗
˜T D 11
(j) ΓOF =
(j)
0
0
∗
−Ir
0
∗
∗
∗
−Ir
∗
∗
∗
∗
(20)
ΓOF (3, 6) (j)
˜ (j) H C 1 ˜H W
(j)
ΓOF (6, 6)
¯ where j = 1, 2, ...L, (j) ΓOF (2, 2) = −P (j) + αA˜(j) H ˜ (j) Y + αH T A˜(j)T + αY T B ˜ (j)T , +αB (j) T ˜ (j)T T ˆ (j)T ΓOF (2, 3) = αH F +Y B , (j) T (j) (j) ˜ ˜ ΓOF (2, 6) = −αH + A H + B Y (j) ˆ (j) Y, Γ (3, 6) = F˜ (j) H + B
(17)
where Ψ(j) and Φ(j) are obtained from (15b,c) by assigning for each vertex j the appropriate matrices. We readily find the following vertex-dependent condition for the stochastic uncertain discrete-time case: Corollary 1: Inequality (17) is satisfied iff there exist ¯ G ∈ matrices: 0 < P (j) ∈ Rn×n , ∀j = 1, 2, .., L, Rn×(2n+p+r) and H ∈ Rn×n that satisfy the inequality ∆ Ω(j) < 0 where Ω(j) =
0
Theorem 2: Consider the augmented system (8a,b) and (11) where the original system matrices components ¯j , D ¯ j , j = 1, ...l − 1, lie within the polytope of (2). For a N prescribed scalar γ > 0, and positive tuning scalar α > 0, there exists a dynamic output-feedback controller that 2 achieves negative JE for all nonzero w ∈ ˜lF ([0, ∞); Rp ), k if there exist n1 × n1 matrices ¯ where n1 = l(m + r) − r, a matrix P (j) > 0, j = 1, 2, ..L, H and a matrix Y that satisfy the following set of LMIs:
Inequality (15a) thus follows from the fact that it is the left side (1,1) matrix block in the latter product. In the uncertain case, we assume that the system parame¯ vertices. ters encounter polytopic-type uncertainty with L (j) T (j) ¯ Choosing then V (x, t) = x (t)P x(t), j = 1, 2, .., L, we consider:
(j),T
2 T ˜T ˜w −γ Ip B 0 D 11 (j) ∗ 0 0 −P (j) ˆ = ∗ Ψ ∗ −P (j) 0 ∗ ∗ −Ir ∗
Taking G = [0 αH [0 0 0]] where H is a n × n matrix, the following result is obtained:
where ΨΦ(1,1) = Ψ + ΦP ΦT , ΨΦ(1,2) = −GT − ΦH T + ΦP, ΨΦ(2,2) = −H − H T + P.
Ψ(j) + Φ(j) P (j) Φ(j)T ≤ 0,
Ψ(j) + GT Φ(j)T + Φ(j) G ∗ < 0, −H − H T + P (j) −G + H T Φ(j)T (18)
OF (j)
ΓOF (6, 6) = −H − H T + P (j) . In the latter case the ’state-feedback’ gain is given by: ¯l−1 −A¯l−2 −A¯l−3... −A¯0 = Y H −1 , (21) ¯1... B ¯0 B K= B
from where the system matrices of the actual dynamic output-feedback controller of Figure 2 are obtained.
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Proof: We replace A˜(j) and F˜ (j) of (19c) by A˜(j) + ˆ (j) K, respectively and we denote KH by Y. ˜ (j) K, F˜ (j) + B B Substituting in (19), the LMI of (20) is readily obtained. We note that H in (20) is required to be non-singular. (j) This, however, follows from the fact that ΓOF (6, 6) in (20) must be negative definite in order for (20) to be feasible.
systems is solved without pre-assigning any structure to the Lyapunov function. This is true for both: nominal and uncertain systems. The solutions in this work are obtained via LMIs of large dimensions. These LMIs are extremely sparse and they can thus be readily solved. The simple system in Example 1 is chosen in order to demonstrates the tractability and applicability of our solution method.
5. EXAMPLES
REFERENCES
A. Bouhtouri, D. Hinriechsen, and AJ. Pritchard. H∞ type control for discrete-time stochastic systems. International Journal of Robust and Nonlinear Control, 9:923– We consider the following uncertain system 948, 1999. yk+2 − 1.6yk+1 − (1.2 + a)yk − 0.16νk yk+1 + 0.08νk yk (22) BS. Chen and W. Zhang. Stochastic H∞ control with = uk+1 − 1.2uk , state-dependent noise. IEEE Transections on Automatic Control, 49(1):45–57, 2004. where a is an uncertain parameter that lies in the interval H. Dong, W. Zidong, SX. Ding, and H. Gao. Event-based a ∈ [−0.1 0.1]. It is required to find a dynamic output H∞ filter design for a class of nonlinear time-varying feedback controller that will stabilize the system in the systems with fading channels and multiplicative noises. mean square sense and that minimizing the induced l2 IEEE Transactions on Signal Processing, 63(13):3387– norm of the system closed-loop complementary sensitivity. 3395, 2015. The system is readily transformed to the augmented model V. Dragan and T. Morozan. Mixed input-output optimizaof (8a,b): tion for time-varying ito systems with state dependent noise. Dynamics of Continuous, Discrete and Impulsive 1.6 1.2 + a −1.2 1 1 Systems, 3:317–333, 1997. 1 0 0 ξk + 0 wk + 0 Kξk ξk+1 = V. Dragan and A. Stoica. A γ attenuation problem 0 0 0 1 for discrete-time time-varying stochastic systems with 0 0.16 −0.08 0 multiplicative noise. Reprint series of the Institute of Mathematics of the Romanian Academy, 10:7–13, 1998. 0 0 ξk ν k , + 0 E. Gershon and U. Shaked. H∞ -output-feedback of 0 0 0 discrete-time systems with state-multiplicative noise. Automatica, 44(2):574–579, 2008. where zk = [0 0 − 1.2]ξk + Kξk describes the signal that is fedback to the summation Σ1 in Figure 1 E. Gershon and U. Shaked. Advanced topics in control and estimation of state-multiplicative noisy systems. LNCIS, due to the disturbance w that acts on the bullet that Springer, London, 2013. proceeds this summation and where, because N0 = 1 in the configuration of (9b), N0 K = K is the ’state- E. Gershon, U. Shaked, and I. Yaesh. H∞ control and estimation of state-multiplicative linear systems. Springer, feedback gain’ in the augmented system that produces the London, 2005. dynamic controller parameters according to (14). Solving for the nominal case (i.e a = 0) by applying the result of M. Green and DJN. Limebeer. Linear Robust Control. Prentice Hall, New Jersy, 1995. Theorem 1, a near minimum attenuation level of γ = 3.712 was obtained for the controller whose transfer function is J. Hu, Z. Wang, B. Shen, and H. Gao. Quantised recursive filtering for a class of nonlinear systems with multiplicaGc (z) = 2.88z+1.641 z−1.641 . tive noises and missing measurements. International In the uncertain case, solving the problem by applying the Journal of Control, 86(4):650–663, 2013. ’quadratic’ result, an attenuation level of γ = 5.77 was T. Kailath. Linear Systems. Prentice-Hall, New Jersy, obtained. The transfer function of the resulting controller 1980. is Gc (z) = 2.8z+1.661 X. Li and H. Gao. A new model model transformation z−1.661 . Applying next the result of Theorem 2 which yields the vertex dependent Lyapunov solution, an of discrete-time systems with time-varying delay and its attenuation level of γ = 5.7318 was obtained for α = 20 application to stability analysis. IEEE Transections on . The stability gain with the controller Gc (z) = 2.8z+1.657 Automatic Control, 56(9):2072–2178, 2011a. z−1.659 and phase margins achieved by the latter controller, for X. Li and H. Gao. A unified approach to the stability of generalized static neural networks with linear fractional a = 0 and νk ≡ 0, are 1.33 and 14.850 , respectively. uncertainties and delays. IEEE Transections on Systems and Cybernetics, 41(5):1275–1286, 2011b. 6. CONCLUSION MC. Oliveira and RE. Skelton. Stability test for constrained linear systems. Springer, Lecture Notes in The problem of dynamic H∞ output-feedback control of Control and Information Sciences, London, 2001. discrete-time linear systems with multiplicative stochastic uncertainties is solved. Solutions are obtained for both C. Scherer and S. Weiland. Linear Matrix Inequalities in Control. e-book, Holand, 2006. nominal and uncertain polytopic-type systems. In the latter case a less conservative approach is presented which V. Suplin and U. Shaked. Robust H∞ output-feedback control of linear discrete-time systems. System and applies a vertex-dependent Lyapunov function. The main Control Letters, 54:799–808, 2005. merit of this work is that, for the first time, the problem of the dynamic output-feedback control for stochastic 5.1 Output-feedback control
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¯0 δ l−1N
... ¯ δN l−2
↓ wk ↓ • •• 1 −
¯ N l−1
↑
↑
yk+l−1 δ
−1
yk+l−2 δ
↓ ¯ l−1+νk Fl−1 D
yk
...
−1
δ
↓ ¯ l−2+νk Fl−2 D
−1
→ ↓ ¯ +νk F D 0 0
uk+l−3
•
↓
δ −1
...
...
uk+l−2
δ −1
δ −1
↓
A¯0
A¯l−3
↓ ¯0 δ l−1B
... uk
•
←
A¯l−2
2 ← −↑
¯ δB l−2 ¯ B l−1
... Fig. 1: The closed-loop fractional description
yk
• ↓
¯0 B
←
... ¯ B l−2
↓
u•k
δ −1
...uk+l−3
δ −1
↓
↓
A¯l−3
A¯0
uk+l−2 ↓ A¯l−2
... Fig. 2: The structure of the controller
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←
δ −1
← 2 −↑
← ¯ B l−1
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 3823–3828 State-multiplicative Noisy Systems: Robust State-multiplicative Noisy H∞ Output-Feedback Control Robust State-multiplicative Noisy Systems: Systems: Robust H Output-Feedback Control ∞ H∞ Output-Feedback Control ∗,∗∗ ∗∗
Eli Gershon Uri Shaked ∗∗ Eli Gershon ∗,∗∗ ∗,∗∗ Uri Shaked ∗∗ Eli Gershon Uri Shaked ∗ Holon Institute of Technology, Medical Engineering, Holon, Israel ∗ ∗∗ Institute of Technology, Medical Engineering, Holon, Israel School of Electrical Engineering, Tel-Aviv University, Tel-Aviv ∗ Holon Holon Institute ofIsrael Technology, Medical Engineering, Holon, Israel ∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, (e-mail: gershon@ eng.tau.ac.il) ∗∗ School 69978, of Electrical Engineering, Tel-Aviv University, Tel-Aviv Israel (e-mail: gershon@ eng.tau.ac.il) 69978, Israel (e-mail: gershon@ eng.tau.ac.il) Abstract: Linear discrete-time systems with stochastic and deterministic polytopic type Abstract: systems stochastic and deterministic polytopic type uncertaintiesLinear in theirdiscrete-time state-space model arewith considered. A dynamic output-feedback controller Abstract: Linear discrete-time systems stochastic and deterministic polytopic type uncertainties in atheir state-space model arewith considered. A of dynamic output-feedback controller is obtained via new approach which allows a derivation a controller in spite of parameter uncertainties in their state-space model are considered. A dynamic output-feedback controller is obtained via newproposed approachapproach which allows derivation of a controller spite of parameter uncertainty. In athe the asystem is described via aindifference equation is obtained via new approach which asystem derivation of a controller spite ofparameters. parameter uncertainty. In athe proposed approach described via controller aindifference equation and an augmented system is then usedallows tothe obtain the isoutput-feedback uncertainty. In isthe proposed approach system described tovia a quadratic difference equation and augmented system is then used tothe obtain the isoutput-feedback parameters. The an controller obtained without assuming a specific structure thecontroller Lyapunov and an augmented system is then to obtain the output-feedback parameters. The controller obtained without assuming a specific structure is toobtained thecontroller quadratic Lyapunov function and it is the first time thatused an output-feedback controller for robust stateThe controller is obtained without a the specific structure toobtained the quadratic Lyapunov function and itsystems. is the first that assuming an output-feedback controller is for robust statemultiplicative Thetime controller minimizes stochastic l2 −gain of the closed-loop where function and itsystems. is defined the first thatexpected an output-feedback is obtained for robust statemultiplicative The controller minimizes theofstochastic l2 −gain the closed-loop a cost function is totime be the value the controller standard H performance indexwhere with ∞ of multiplicative systems. The controller minimizes the stochastic l −gain of the closed-loop where 2 a cost function is defined uncertainty. to be the expected value of standard H∞ performance indexofwith respect to the stochastic An example is the given that demonstrates the merit the a cost function is defined uncertainty. to be the expected value of standard H∞ performance indexofwith respect to the stochastic An example is the given that demonstrates the merit the theory. respect theory. to the stochastic uncertainty. An example is given that demonstrates the merit of the theory. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Stochastic H∞ output-feedback control, vertex-dependent Lyapunov function, Keywords: Stochastic multiplicative noise. H∞ output-feedback control, vertex-dependent Lyapunov function, Keywords: Stochastic multiplicative noise. H∞ output-feedback control, vertex-dependent Lyapunov function, multiplicative noise. 1. INTRODUCTION shon and Shaked [2008]. The solution in Dragan and Stoica 1. INTRODUCTION shon Shakedthe [2008]. Theand solution in Dragan andhorizon Stoica [1998]and includes finite the infinite-time 1. INTRODUCTION shon and Shaked [2008]. Theand solution in Dragan andhorizon Stoica [1998] includes the finite the infinite-time problems without transients. One drawback of Dragan and The field of stochastic H∞ control and estimation of state[1998] includes the finite and the infinite-time horizon problems without transients. One drawback of Dragan and Stoica [1998] is the fact that in the infinite-time horizon The field of stochastic control and estimation of statemultiplicative systemsHhas been developed following the ∞ problems without transients. One drawback of Dragan and control andtheory estimation stateThe fieldin of the stochastic Hhas ∞H is number the factofthat in the infinite-time horizon case, an[1998] infinite Linear Matrix Inequality (LMI) multiplicative systems been developed following the advance stochastic that of was de- Stoica 2 control Stoica [1998] is the fact that in the infinite-time horizon multiplicative systems has been developed following the case, an infinite number of Linear Matrix Inequality (LMI) sets should be solved. We also note that the fact that advance in inthe theearly stochastic H270’s control theory that developed 60’s and (see Gershon et al.was [2005] an infinite of Linear Matrix Inequality (LMI) advance inthe theearly stochastic H270’s theory that de- case, sets should be number solved. We also that the fact that in Dragan and Stoica [1998] thenote measurement coupling veloped in 60’s and (seeemerged, Gershon et aal.was [2005] for a comprehensive study). Itcontrol has as natural sets should be solved. We also note that the fact that veloped in the early 60’s and 70’s (see Gershon et al. [2005] Dragan andaccommodate Stoica [1998]any theuncertainty measurement coupling matrix can not is a practical for a comprehensive It progress has emerged, natural extension, in parallelstudy). with the madeastoathe field in in Dragan and Stoica [1998] the measurement coupling for a comprehensive study). It has emerged, as a natural can not anywhere uncertainty is a practical restriction, for accommodate example in cases the measurements extension, in parallel with the progress made field matrix of deterministic H∞ control theory, drawing onto thethe advent not accommodate any uncertainty is a practical extension, in field parallel with the progress made the field matrix restriction, forderivatives example in(e.g. cases where the measurements include can state acceleration control of an deterministic H∞tocontrol theory, drawing onto the advent of the latter cope with MIMO control problems restriction, for example in cases where the measurements of deterministic H control theory, drawing on the advent derivatives acceleration control et of al. an aircraft state or missile). The (e.g. treatment of Bouhtouri of the and latterLimebeer field ∞to cope control Green [1995]with and MIMO with the robustproblems control include include state derivatives acceleration control of al. an of latterLimebeer field to cope with control problems or missile). The (e.g. treatment Bouhtouri et [1999] includes the derivation of the of stochastic Bounded Green and [1995] and MIMO withWeiland the robust control andthe estimation problems, Scherer and [2006]. This aircraft aircraft or missile). The treatment of Bouhtouri et al. Green and Limebeer [1995] and with the robust control includes the derivation of the Bounded Real Lemma (BRL) and concerns onlystochastic the stationary case and problems, and Weiland fieldestimation is recognized now asScherer a central sub-field [2006]. withinThis the [1999] includes the nonlinear derivation of the Bounded and estimation and Weiland [2006]. This Real (BRL) and concerns onlystochastic thewere stationary case whereLemma two coupled inequalities obtained. field is of recognized now asScherer acontrol central sub-field within the [1999] theory linearproblems, stochastic Dragan and Morozan Real Lemma (BRL) and concerns only the stationary case field is of recognized now as[2004] acontrol central sub-field theory linear andwithin Morozan [1997], Chen andstochastic Zhang (seeDragan also Gershon et the al. where two coupled nonlinear inequalities were obtained. In Gershon and Shaked [2008] the solution of the outputtheory of control Dragan and Morozan [1997], Chen andstochastic Zhang [2004] (see also Gershon et al. where two coupled nonlinear inequalities were obtained. [2005] for alinear comprehensive review) and nonlinear stochasIn Gershon and Shaked the obtained solution of outputfeedback control problem[2008] has been bythe transform[1997], Chen and Zhang [2004] (see also Gershon et al. [2005] for a (see, comprehensive review) nonlinear stochastic systems for example, Dongand et al. [2015],Hu et al. In Gershon and to Shaked [2008] theInobtained solution of the outputfeedback control problem has been by transforming the problem one of filtering. the finite horizon case [2005] for a (see, comprehensive review) nonlinear stochastic systems forsolution example, Dongand et al. [2015],Hu et al. [2013]). Numerous methods have been applied to feedback control problem has been obtained by transforming the problem oneobtained of filtering. In via theDLMIs finite horizon case the solution has to been there [Difference tic systems (see, for example, Dong et al. [2015],Hu et al. [2013]). Numerous methods have applied to ing the problem to one of filtering. In the finite horizon case various control and solution estimation patterns (seebeen Gershon et al. the solution hasInequalities] been obtained there via DLMIs [Difference Linear Matrix whereas in the stationary case [2013]). Numerous solution methods have been applied to various control and estimation patterns (seethe Gershon et al. the solution has been obtained there via DLMIs [Difference [2005] and the references therein) and over last decade Matrix whereas in the stationary case the solution hasInequalities] been obtained by assigning a special strucvarious control and patterns (see Gershon et al. Linear [2005] and the references andtoover the last decade the latter field has estimation been therein) expanded include stochastic Linear Matrix Inequalities] whereas inthereby the stationary case the solution has been obtained by assigning a special structure to the Lyapunov matrix solution, imposing an [2005] and thetime references last decade the latter field hasdelay beenoftherein) expanded toover include stochastic systems with various and kinds (i.ethe constant time- the solution has been obtained by assigning a special structure to the Lyapunov matrix The solution, thereby imposing over-design on the solution. robust case could not an be the latter field has been expanded to include stochastic systemsslow withand timefast delay of various kinds time- ture to the Lyapunov matrix solution, thereby imposing an delay, varying delay) in (i.e bothconstant the nominal over-design on the solution. The robust case could notfact be treated in Gershon and Shaked [2008] owing to the systems with time delay of various kinds (i.e constant timedelay, and fast varying delay) inand both the nominal and theslow uncertain cases (see Gershon Shaked [2013], over-design on obtained the solution. The robust caseincould notfact be treated in Gershon and Shaked [2008] owing to the that the LMI there is not affine the system delay, and Gao fast varying delay) both the nominal and theslow uncertain cases (see and Gershon and Shaked [2013], see also Li and [2011a] Li in and Gao [2011b] for treated in Gershon and Shaked [2008] owing to the fact that the LMI is not an affine in thedynamic system matrices. The obtained problem there of finding optimal and the uncertain cases (see and Gershon andGao Shaked [2013], see Li and Gao [2011a] Li and [2011b] for that the LMI obtained there is not affine in the system the also deterministic case). matrices. The problem of finding an optimal output-feedback for state-multiplicative systems dynamic has thus see also Li and Gao [2011a] and Li and Gao [2011b] for the deterministic case). matrices. The problem of finding an optimal dynamic output-feedback for state-multiplicative systems has used thus been constrained in the past by the solution method the deterministic case). output-feedback for state-multiplicative systems has used thus been constrained in the past by the solution method and conservative solutions have then been obtained. The problem of H∞ dynamic output-feedback has always been constrained in the past by the solution method used The problem of H output-feedback has In always been a central issue in modern control theory. the and conservative solutions have then been obtained. ∞ dynamic In present manuscript solve thebeen problem of discreteandthe conservative solutionswe have then obtained. The problem of H output-feedback always ∞ dynamic been a central issue incase, modern control theory. In the stochastic discrete-time this problem was has tackled by In the present manuscript we solve the problem of discretetime dynamic output-feedback control via a new approach been central issue inDragan modern control theory. InGerthe stochastic discrete-time case, this was tackled by In the present manuscript we solve the problem of discretevariousa research groups andproblem Stoica [1998] and time dynamicusoutput-feedback via athe newproblem approach that enables to solve, for thecontrol first time, of stochastic discrete-time case, this problem was tackled by various research groups Dragan and Stoica [1998] and Ger- time dynamic output-feedback control via a new approach that enables us to solve, for the first time, the problem of measurement control of uncertain systems without the re various research groups Dragan and Stoica [1998] and GerThis work was supported by the Israeli Science Foundation no. that enables to solve, for the first the problem of measurement control of uncertain systems without the restrictions thatus have been indicated in time, the above mentioned 737/2011. This work was supported by the Israeli Science Foundation no. measurement control of uncertain systems without the re This work was supported by the Israeli Science Foundation no. strictions that have been indicated in the above mentioned 737/2011. strictions that have been indicated in the above mentioned 737/2011.
Copyright © 2017 IFAC 3886 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2017 responsibility IFAC 3886Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 3886 10.1016/j.ifacol.2017.08.488
Proceedings of the 20th IFAC World Congress 3824 Eli Gershon et al. / IFAC PapersOnLine 50-1 (2017) 3823–3828 Toulouse, France, July 9-14, 2017
works. Applying a linear fractional approach, we apply the method of Suplin and Shaked [2005] and transform the problem of finding a dynamic controller to one of finding a state-feedback controller gain matrix. We are thus able to treat two control design cases which are fundamental in control engineering on one hand and, on the other hand, we are also able to cope with the robust version of the latter two control patterns, without pre-assigning structure to the relevant Lyapunov matrix function solution. In the uncertain case, we bring a vertex-dependent solution (in contrast to the traditional ‘quadratic‘ solution). The paper is organized as follows: starting with the difference equation representation, we first introduce in Section 2 the linear fractional intermediate description of the system. In Section 3, we consider the stabilization and the l2 −gain of the nominal closed-loop system. Based on the result of Section 3, the solution of the robust dynamic output-feedback control problem is given in Section 4 for the case where the system encounters polytopictype parameter uncertainty. Two different control solution methods are applied there: the quadratic and the vertex dependent approaches. In Section 5, an example is given that demonstrates the applicability and tractability of our method we developed. Notation: Throughout the paper the superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space, Rn×m is the set of all n × m real matrices, N is the set of natural numbers and the notation P > 0, (respectively, P ≥ 0) for P ∈ Rn×n means that P is symmetric and positive definite (respectively, semidefinite). We denote by L2 (Ω, Rn ) the space of squareintegrable Rn − valued functions on the probability space (Ω, F, P), where Ω is the sample space, F is a σ algebra of a subset of Ω called events and P is the probability measure on F . By (Fk )k∈N we denote an increasing family of σ-algebras Fk ⊂ F. We also denote by ˜l2 (N ; Rn ) the ndimensional space of nonanticipative stochastic processes {fk }k∈N with respect to (Fk )k∈N where fk ∈ L2 (Ω, Rn ). On the latter space the following l2 -norm is defined: ∞ ∞ ||fk ||2 } = E{||fk ||2 } < ∞, ||{fk }||˜2l = E{
where
¯ i = 1, ..., L,
(3)
¯ is the number of vertices. and where L We seek an output-feedback dynamical controller of the type l−2 l−1 ¯l−1−i yk+i uk+l−1 + (4) A¯i uk+i = B i=0
i=0
that, over the polytope Ω stabilizes the system exponentially in the mean-square sense and minimizes the induced l2 −norm of the resulting closed-loop.
The above system is described (for clarity of representation we take Fi = 0, i = l, ..., 2l − 1) in the block diagram of Figure 1 where δ denotes the one unit time shift operator. We represent the above system by the augmented observability canonical form (see Kailath [1980]) as follows: we ∆ define the following augmented state-vector in n , n = l(m + r) − r, T T T (5) ξk = yk+l−1 . . . ykT uTk+l−2 . . . uTk , yk+l−2
and obtain ˜ k + νk F˜ ξk + (B ˜ + νk B)Kξ ˆ ξk+1 = Aξ k where
2
0
¯l−1, i ¯0, i ... N N ¯ l−1, i , ¯ 0, i ... D Ωi = D F0, i ... F2l−2, i
0
{fk } ∈ ˜l2 (N ; Rn ),
where || · || is the standard Euclidean norm.
−D¯ l−1
˜ A=
Im 0 . 0 0 0 0 . 0
−Fl−1
F˜ =
0 0 . 0 0 0 0 . 0
. 0 Im . . . . . . . . 0 0 . . . . . . .
¯0 N ¯l−1 ¯1 . . . N . . −D . . 0 0 . . . 0 . . 0 0 . . . 0 . . . . . . . . . Im 0 0 . . . 0 , . . 0 0 . . . 0 . . 0 Ir 0 . . 0 . . 0 0 Ir . . 0 . . . . . . . . . . 0 0 . . Ir 0
. . . . . . . . . .
. −F0 Fl+1 . 0 0 . 0 0 . . . 0 0 0 . 0 0 . 0 0 . 0 0 . . . . 0 0
. . . . . . 0 0 . .
. . . . . . . . . .
(7)
. F2l−1 . 0 . 0 . . . 0 , . 0 . 0 . 0 . . 0 0
T ˆ = F T 0. . . 0 0 T ˜= N ¯0 0. . . 0 Ir 0 . . . 0 T , B B l
2. PROBLEM FORMULATION
(6)
(7) We consider the r-inputs m-outputs system that is de 2l−1 ] = and K=[K1 K2 , ..., Kl Kl+1 , ..., K scribed by the following difference equation: ¯0 B ¯l−1−A¯l−2 −A¯l−3 . . .−A¯0 . ¯1. . .B B l−1 l−1 ¯ i + Fi νk )yk+i = ¯l−1−i +F2l−1−i νk )uk+i (1) We note that the above state space model is non minimal, (D (N yk+l + for both the nominal and the uncertain cases. It is already i=0 i=0 observable but may be uncontrollable due to the additional m r where yk ∈ R is the system output, uk ∈ R is the roots added to the augmented system. A ’state-feedback’ control input, {νk } is a standard zero-mean real scalar from the state vector ξ in this model to the ’control input’ k white-noise sequence with E{νk νj } = δkj and where u will produce the components Ai and Bi of the k+l−1 ¯i ∈ Rm×r , Fi ∈ Rm×r , i = l, ..., 2l − ¯ i ∈ Rm×m , N D controller (4) that relates y k with uk . 1 and Fi ∈ Rm×m , i = l, ..., l − 1. We assume that system is detectable and stabilizable. We further assume that the In the stochastic H∞ control setting, we obtain the followparameters in (1) are not precisely known and that they ing state-space representation: ˜ k+B ˜w wk + BKξ ˜ k + νk (F˜ + BK)ξ ˆ reside in the following polytope, ξk+1 = Aξ k, (8) ˜ 11 wk , ˜ ξk + D Ω = Co{Ω1 , Ω2 , ..., ΩL¯ } , (2) zk = C˜1 ξk + νk W 3887
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˜ and B ˆ depend on the way by which νk affects where W ¯ Ni , i = 1, 2, ..., N. In our model, the disturbance acts on yk+l with
∆ ΓP =
¯1 ...N ¯l−1 ] + N ¯0 K. (9) ˜w = [Im 0 0 ...0]T and C˜1 = [0...0 N B ˜ 11 depends on the control problem to be The matrix D solved. It describes the way that the disturbance affects the objective function. In the following we describe two problems in which the disturbance effect is considered differently: Problem 1: Sensitivity minimization The sensitivity matrix function is the operator that relates the input {wk } in Figure 1 with the signal that is measured at the bullet that immediately proceeds the disturbance entry point. The minimization of the H∞ norm of the ˜ 11 = 0 sensitivity function is thus achieved by taking D in (8b). Problem 2: Complementary sensitivity minimization The complementary sensitivity is the operator that relates the input {wk } in Figure 1 with the signal that precedes the entry point. In our case, we seek to reduce the l2 −gain ˜ 11 = 0. of the latter transference. We thus require that D We note that in our system the measurement signal is not contaminated by sensor noise. However, the complementary sensitivity function is also the transference (noting that the phase is reversed) from the traditional entry point of the measurement noise to the output measurement signal. By minimizing the induced L2 norm of this function one also minimizes the effect of measurement noise if it exists.
3. THE H∞ CONTROLLER FOR NOMINAL SYSTEMS In order to solve the last two minimization problems for the augmented system we first bring the stochastic BRL result that was obtained in (see Gershon et al. [2005], p.104). This result refers to the following n-th order system xk+1 = (A + F νk )xk + B1 wk , zk = (C1 + W νk )xk + D11 wk ,
xi = 0, i ≤ 0,
(10)
0
P AT P F T P C1T P W T
∗ −γ 2 Ip B1T
∗
T D11
0
0
0
0
∗
−P
0
0 0
∗
∗
−Ir
∗
−P
∗
∗
∗
∗
∗
∗
∗
0
∗
∗
−Ir
< 0. (12)
Remark 1: Lemma 1 has been used in the past to solve the output feedback control problem, Gershon et al. [2005], Gershon and Shaked [2008]. Unfortunately, when seeking a dynamical output-feedback controller, the resulting inequalities become BMIs [Bilinear Matrix Inequalities]. To circumvent this difficulty, a special structure had to be assigned to the matrix P - a procedure which led to considerable overdesign. Based on Lemma 1, the ’state-feedback controller’ can be readily derived for the augmented system by applying the latter result to the system of (8a,b). In this way no special assumption on the structure of P is required. We thus obtain the following result: Theorem 1 Consider the augmented system (8a,b) and (11). For a prescribed scalar γ > 0, there exists a statefeedback gain that achieves negative JE for all nonzero 2 wk ∈ ˜lF ([0, ∞); Rp ), if there exist n1 × n1 matrix P > 0, k where n1 = l(m + r) − r and a matrix Y that satisfy the following LMI condition: −P 0 P A˜T + Y T B˜ T P F˜ T + Y T Bˆ T P C˜ T P W ˜T 1
0
˜T D 11
0
−P
0
0
0
∗
∗
−P
0
0
∗
∗
∗
−Ir
0
∗
∗
∗
∗
−Ir
∗
2
−γ Ip
˜T B w
∗
∗
∗ ∗ ∗
< 0.(13)
In the latter case, the ’state-feedback gain’ is given by: ¯0 B ¯l−1 −A¯l−2 −A¯l−3... −A¯0 = Y P −1 , (14) ¯1... B K= B
from where the system matrices of the actual output feedback controller of Figure 2 are obtained. Proof We replace A, B1 and C1 , W, D11 of (10a,b) by ˆ ˜w and C˜1 , W ˜, D ˜ 11 of (8a,b) respectively, and A˜ + BK, B we denote KP by Y. Substituting in (12) the LMI of (13) is readily obtained.
with the index of performance:
4. THE H∞ CONTROLLER FOR UNCERTAIN SYSTEMS
∆
JE = ||zk ||˜2l − γ 2 ||wk ||˜2l , 2
−P
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2
(11)
where wk ∈ Rp , zk ∈ Rr . The following has been obtained in Dragan and Stoica [1998],Gershon et al. [2005]: Lemma 1 Consider the system (10a,b) and the above JE . The system is exponentially stable in the mean square sense and, for a prescribed scalar γ > 0, the requirement of 2 JE < 0 is achieved for all nonzero wk ∈ ˜lF ([0, ∞); Rp ), iff k there exists n×n matrix P > 0, that satisfies the following inequality:
The main advantage of our approach is the ability it provides to obtain a solution to the output-feedback control problem in the case of polytopic uncertain parameters. Two possible solution methods can be considered. The first produces a ’quadratic’ solution where a single Lyapunov function is considered over the whole uncertainty interval. This is readily obtained by applying the result of Theorem 1 to all the vertices of the uncertainty polytope applying the same matrices P and Y. The second solution method is based on the fact that a less conservative solution may
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be obtained by considering vertex-dependent Lyapunov functions over the whole uncertainty polytope. In order to apply the latter approach, we first present the vertexdependent solution of the BRL of Lemma 1. We consider the system of (10a,b) and the LMI of (12). Applying Schur’s complements to the latter LMI, we obtain T
where
Ψ + ΦP Φ ≤ 0,
Ψ=
−γ 2 Ip B1T
0
0
∗
−P
0
0
∗
∗
−P
0
∗
∗
∗
−Ir
∗
∗
∗
∗
0
0
A = F . , Φ 0 0 C1 0
−Ir
(15)
¯ ∀j = 1, 2, .., L. Applying the above result, the solution of the vertexdependent robust BRL problem can be readily derived for the augmented system. Considering the augmented system (8a,b) where K = 0 and (11), where the system matrices components lie within the polytope of (2), we obtain the following condition: (j) ˆ +GT Φ(j)T +Φ(j) G −GT +Φ(j) H ∆ Ψ ˆ (j) = < 0, Ω ∗ −H −H T +P (j)
W
The structure of (15) can be used in order to reduce the conservatism entailed by applying the Lyapunov function uniformly over the uncertainty polytope. However, the standard application of the latter method in Oliveira and Skelton [2001] is not readily applicable to the stochastic case where taking the adjoint of the system, instead of the original one, is theoretically unjustified. We therefore bring below a vertex-dependent method that is more suitable and readily applicable to the uncertain stochastic case. We start with (15) and obtain the following lemma: Lemma 2: Inequality (15a) is satisfied iff there exist matrices: 0 < P ∈ Rn×n , G ∈ Rn×(2n+p+r) and H ∈ Rn×n that satisfy the following inequality T T −GT + ΦH ∆ Ψ + G Φ + ΦG Ω= < 0. (16) ∗ −H − H T + P Proof: Substituting G = 0 and H = P in (16), inequality (15a) is obtained. To show that (16) leads to (15a) we consider I 0 ΨΦ(1,1) ΨΦ(1,2) I Φ = , Ω T ∗ ΨΦ(2,2) 0 I Φ I
Φ
¯ ∀j = 1, 2, ..., L,
0 , 0 0
(19)
∗ ∗ ∗ ∗ −Ir (j)T ˜ T ˜ (j)T ˜ T ¯ ˜ = [0 A F C1 W ], ∀j = 1, 2, ...L.
(j)
−γ 2 I
∗
∗
ΓOF < 0, where p
˜T B w
0
0
0
(j) (j) (j) T ˜ (j)T T ˜ T αH W ΓOF (2, 6) ΓOF (2, 2) ΓOF (2, 3) αH C 1
∗
−P
∗
∗ ∗
∗
˜T D 11
(j) ΓOF =
(j)
0
0
∗
−Ir
0
∗
∗
∗
−Ir
∗
∗
∗
∗
(20)
ΓOF (3, 6) (j)
˜ (j) H C 1 ˜H W
(j)
ΓOF (6, 6)
¯ where j = 1, 2, ...L, (j) ΓOF (2, 2) = −P (j) + αA˜(j) H ˜ (j) Y + αH T A˜(j)T + αY T B ˜ (j)T , +αB (j) T ˜ (j)T T ˆ (j)T ΓOF (2, 3) = αH F +Y B , (j) T (j) (j) ˜ ˜ ΓOF (2, 6) = −αH + A H + B Y (j) ˆ (j) Y, Γ (3, 6) = F˜ (j) H + B
(17)
where Ψ(j) and Φ(j) are obtained from (15b,c) by assigning for each vertex j the appropriate matrices. We readily find the following vertex-dependent condition for the stochastic uncertain discrete-time case: Corollary 1: Inequality (17) is satisfied iff there exist ¯ G ∈ matrices: 0 < P (j) ∈ Rn×n , ∀j = 1, 2, .., L, Rn×(2n+p+r) and H ∈ Rn×n that satisfy the inequality ∆ Ω(j) < 0 where Ω(j) =
0
Theorem 2: Consider the augmented system (8a,b) and (11) where the original system matrices components ¯j , D ¯ j , j = 1, ...l − 1, lie within the polytope of (2). For a N prescribed scalar γ > 0, and positive tuning scalar α > 0, there exists a dynamic output-feedback controller that 2 achieves negative JE for all nonzero w ∈ ˜lF ([0, ∞); Rp ), k if there exist n1 × n1 matrices ¯ where n1 = l(m + r) − r, a matrix P (j) > 0, j = 1, 2, ..L, H and a matrix Y that satisfy the following set of LMIs:
Inequality (15a) thus follows from the fact that it is the left side (1,1) matrix block in the latter product. In the uncertain case, we assume that the system parame¯ vertices. ters encounter polytopic-type uncertainty with L (j) T (j) ¯ Choosing then V (x, t) = x (t)P x(t), j = 1, 2, .., L, we consider:
(j),T
2 T ˜T ˜w −γ Ip B 0 D 11 (j) ∗ 0 0 −P (j) ˆ = ∗ Ψ ∗ −P (j) 0 ∗ ∗ −Ir ∗
Taking G = [0 αH [0 0 0]] where H is a n × n matrix, the following result is obtained:
where ΨΦ(1,1) = Ψ + ΦP ΦT , ΨΦ(1,2) = −GT − ΦH T + ΦP, ΨΦ(2,2) = −H − H T + P.
Ψ(j) + Φ(j) P (j) Φ(j)T ≤ 0,
Ψ(j) + GT Φ(j)T + Φ(j) G ∗ < 0, −H − H T + P (j) −G + H T Φ(j)T (18)
OF (j)
ΓOF (6, 6) = −H − H T + P (j) . In the latter case the ’state-feedback’ gain is given by: ¯l−1 −A¯l−2 −A¯l−3... −A¯0 = Y H −1 , (21) ¯1... B ¯0 B K= B
from where the system matrices of the actual dynamic output-feedback controller of Figure 2 are obtained.
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Proof: We replace A˜(j) and F˜ (j) of (19c) by A˜(j) + ˆ (j) K, respectively and we denote KH by Y. ˜ (j) K, F˜ (j) + B B Substituting in (19), the LMI of (20) is readily obtained. We note that H in (20) is required to be non-singular. (j) This, however, follows from the fact that ΓOF (6, 6) in (20) must be negative definite in order for (20) to be feasible.
systems is solved without pre-assigning any structure to the Lyapunov function. This is true for both: nominal and uncertain systems. The solutions in this work are obtained via LMIs of large dimensions. These LMIs are extremely sparse and they can thus be readily solved. The simple system in Example 1 is chosen in order to demonstrates the tractability and applicability of our solution method.
5. EXAMPLES
REFERENCES
A. Bouhtouri, D. Hinriechsen, and AJ. Pritchard. H∞ type control for discrete-time stochastic systems. International Journal of Robust and Nonlinear Control, 9:923– We consider the following uncertain system 948, 1999. yk+2 − 1.6yk+1 − (1.2 + a)yk − 0.16νk yk+1 + 0.08νk yk (22) BS. Chen and W. Zhang. Stochastic H∞ control with = uk+1 − 1.2uk , state-dependent noise. IEEE Transections on Automatic Control, 49(1):45–57, 2004. where a is an uncertain parameter that lies in the interval H. Dong, W. Zidong, SX. Ding, and H. Gao. Event-based a ∈ [−0.1 0.1]. It is required to find a dynamic output H∞ filter design for a class of nonlinear time-varying feedback controller that will stabilize the system in the systems with fading channels and multiplicative noises. mean square sense and that minimizing the induced l2 IEEE Transactions on Signal Processing, 63(13):3387– norm of the system closed-loop complementary sensitivity. 3395, 2015. The system is readily transformed to the augmented model V. Dragan and T. Morozan. Mixed input-output optimizaof (8a,b): tion for time-varying ito systems with state dependent noise. Dynamics of Continuous, Discrete and Impulsive 1.6 1.2 + a −1.2 1 1 Systems, 3:317–333, 1997. 1 0 0 ξk + 0 wk + 0 Kξk ξk+1 = V. Dragan and A. Stoica. A γ attenuation problem 0 0 0 1 for discrete-time time-varying stochastic systems with 0 0.16 −0.08 0 multiplicative noise. Reprint series of the Institute of Mathematics of the Romanian Academy, 10:7–13, 1998. 0 0 ξk ν k , + 0 E. Gershon and U. Shaked. H∞ -output-feedback of 0 0 0 discrete-time systems with state-multiplicative noise. Automatica, 44(2):574–579, 2008. where zk = [0 0 − 1.2]ξk + Kξk describes the signal that is fedback to the summation Σ1 in Figure 1 E. Gershon and U. Shaked. Advanced topics in control and estimation of state-multiplicative noisy systems. LNCIS, due to the disturbance w that acts on the bullet that Springer, London, 2013. proceeds this summation and where, because N0 = 1 in the configuration of (9b), N0 K = K is the ’state- E. Gershon, U. Shaked, and I. Yaesh. H∞ control and estimation of state-multiplicative linear systems. Springer, feedback gain’ in the augmented system that produces the London, 2005. dynamic controller parameters according to (14). Solving for the nominal case (i.e a = 0) by applying the result of M. Green and DJN. Limebeer. Linear Robust Control. Prentice Hall, New Jersy, 1995. Theorem 1, a near minimum attenuation level of γ = 3.712 was obtained for the controller whose transfer function is J. Hu, Z. Wang, B. Shen, and H. Gao. Quantised recursive filtering for a class of nonlinear systems with multiplicaGc (z) = 2.88z+1.641 z−1.641 . tive noises and missing measurements. International In the uncertain case, solving the problem by applying the Journal of Control, 86(4):650–663, 2013. ’quadratic’ result, an attenuation level of γ = 5.77 was T. Kailath. Linear Systems. Prentice-Hall, New Jersy, obtained. The transfer function of the resulting controller 1980. is Gc (z) = 2.8z+1.661 X. Li and H. Gao. A new model model transformation z−1.661 . Applying next the result of Theorem 2 which yields the vertex dependent Lyapunov solution, an of discrete-time systems with time-varying delay and its attenuation level of γ = 5.7318 was obtained for α = 20 application to stability analysis. IEEE Transections on . The stability gain with the controller Gc (z) = 2.8z+1.657 Automatic Control, 56(9):2072–2178, 2011a. z−1.659 and phase margins achieved by the latter controller, for X. Li and H. Gao. A unified approach to the stability of generalized static neural networks with linear fractional a = 0 and νk ≡ 0, are 1.33 and 14.850 , respectively. uncertainties and delays. IEEE Transections on Systems and Cybernetics, 41(5):1275–1286, 2011b. 6. CONCLUSION MC. Oliveira and RE. Skelton. Stability test for constrained linear systems. Springer, Lecture Notes in The problem of dynamic H∞ output-feedback control of Control and Information Sciences, London, 2001. discrete-time linear systems with multiplicative stochastic uncertainties is solved. Solutions are obtained for both C. Scherer and S. Weiland. Linear Matrix Inequalities in Control. e-book, Holand, 2006. nominal and uncertain polytopic-type systems. In the latter case a less conservative approach is presented which V. Suplin and U. Shaked. Robust H∞ output-feedback control of linear discrete-time systems. System and applies a vertex-dependent Lyapunov function. The main Control Letters, 54:799–808, 2005. merit of this work is that, for the first time, the problem of the dynamic output-feedback control for stochastic 5.1 Output-feedback control
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¯0 δ l−1N
... ¯ δN l−2
↓ wk ↓ • •• 1 −
¯ N l−1
↑
↑
yk+l−1 δ
−1
yk+l−2 δ
↓ ¯ l−1+νk Fl−1 D
yk
...
−1
δ
↓ ¯ l−2+νk Fl−2 D
−1
→ ↓ ¯ +νk F D 0 0
uk+l−3
•
↓
δ −1
...
...
uk+l−2
δ −1
δ −1
↓
A¯0
A¯l−3
↓ ¯0 δ l−1B
... uk
•
←
A¯l−2
2 ← −↑
¯ δB l−2 ¯ B l−1
... Fig. 1: The closed-loop fractional description
yk
• ↓
¯0 B
←
... ¯ B l−2
↓
u•k
δ −1
...uk+l−3
δ −1
↓
↓
A¯l−3
A¯0
uk+l−2 ↓ A¯l−2
... Fig. 2: The structure of the controller
3891
←
δ −1
← 2 −↑
← ¯ B l−1